On Thursday’s class, we finished Chapter 4 with the Intermediate Value Property. This is a useful property and we could prove it by different ways. On class, Dr. Lecrone showed to us how to use connected sets and continuity to prove this theorem.

We also started talking about differentiation, a topic has close relationship with calculus class. However, as what we learned in class, we noticed that differentiability has a great connection with continuity though, they are not the same. A function is continuous will not imply it is differentiable but if it is differentiable, it is continuous. In order to prove this theorem, we applied Algebraic Limit Theorem again. During the process of this proof, we learned that whenever we use this theorem, we have to make sure that the limit of those functions actually exists! or we assumed that those limits exist.